# Stephan BaierRamakrishna Mission Vivekananda University, Belur Math · Mathematics

Stephan Baier

PhD

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94

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Introduction

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## Publications

Publications (94)

Let p>5 be a fixed prime and assume that α1,α2,α3 are coprime to p. We study the asymptotic behavior of small solutions of congruences of the form α1x12+α2x22+α3x32≡0modq with q=pn, where max{|x1|,|x2|,|x3|}≤N and (x1x2x3,p)=1. (In fact, we consider a smoothed version of this problem.) If α1,α2,α3 are fixed and n→∞, we establish an asymptotic formu...

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence pα when α is a fixed irrational real number and p runs over the primes. In particular, he showed that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has s...

We investigate the distribution of αp modulo one in quadratic number fields 𝕂 with class number one, where p is restricted to prime elements in the ring of integers of 𝕂. Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [On the distribution of αp modulo one in imaginary q...

The distribution of \(\alpha p\) modulo one, where p runs over the rational primes and \(\alpha \) is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents \(\nu >0\) one can establish the infinitude of primes p satisfying \(||\alpha p||\le p^{-\nu }\). The latest record in this regard is Kaisa Matomäki’...

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $p\alpha$ when $\alpha$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||p\alpha||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the dista...

We prove a lower and an upper bound for the large sieve with square moduli in function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in [3] and [6]. Our lower bound in the function field setting contradicts an upper bound obtained in [4]. Indeed, in [5] we pointed out an error in [4].

In 2020, Roger Baker proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribute as expected in arithmetic progressions mod $q$, except for a subset of $\mathcal{S}$ whose car...

We investigate the distribution of $\alpha p$ modulo one in quadratic number fields $\mathbb{K}$ with class number one, where $p$ is restricted to prime elements in the ring of integers of $\mathbb{K}$. Here we improve the relevant exponent $1/4$ obtained by the first and third named authors for imaginary quadratic number fields~\cite{BT} and by th...

In this paper, we establish a version of the large sieve inequality with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude of primes $p$ satisfying $||\alpha p||\le p^{-\nu}$. The latest record in this regard is Kaisa Matom\"aki's lan...

In [3], we derived three results in additive combinatorics for function fields. The proofs of these results depended on a recent bound for the large sieve with sparse sets of moduli for function fields by the first and third-named authors in [1]. Unfortunately, they discovered an error in this paper and demonstrated in [2] that this result cannot h...

Let n be a positive integer and f (x) := x 2 n + 1. In this paper, we study orders of primes dividing products of the form P m,n := f (1)f (2) · · · f (m). We prove that if m > max{10 12 , 4 n+1 }, then there exists a prime divisor p of P m,n such that ord p (P m,n) ≤ n · 2 n−1. For n = 2, we establish that for every positive integer m, there exist...

In [11], the second and third-named authors established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in [3] by the first and second-named authors. In this context, a Central Limit Theorem was established only under a strong hypothesis going beyo...

In this paper, we establish a version of the large sieve with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.

We prove a lower and an upper bound for the large sieve with square moduli for function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in arXiv:1812.05844 by Baier, Lynch and Zhao and arXiv:math/0512271 by Baier and Zhao. Our lower bound in the function field setting contradicts an upper bound...

The second and third-named authors (arXiv:1705.04115) established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in by the first and second-named authors (arXiv:1705.09229). In this context, a Central Limit Theorem was established only under a str...

We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to I. M. Vinogradov, we obtain that the inequality $...

Let $n$ be a positive integer and $f(x) := x^{2^n}+1$. In this paper, we study orders of primes dividing products of the form $P_{m,n}:=f(1)f(2)\cdots f(m)$. We prove that if $m > \max\{10^{12},4^{n+1}\}$, then there exists a prime divisor $p$ of $P_{m,n}$ such that ord$_{p}(P_{m,n} )\leq n\cdot 2^{n-1}$. For $n=2$, we establish that for every posi...

We prove a lower bound for the large sieve with square moduli.

We prove a lower bound for the large sieve with square moduli.

We establish a general large sieve inequality with sparse sets $\mathcal{S}$ of moduli in the Gaussian integers which are in a sense well-distributed in arithmetic progressions. This extends earlier work of S. Baier on the large sieve with sparse sets of moduli. We then use this result to obtain large sieve bounds for the cases when $\mathcal{S}$ c...

Let E be an elliptic curve over \( {\mathbb Q}\). Let p be a prime of good reduction for E. Then, for a prime \( p \ne \ell \), the Frobenius automorphism associated with p (unique up to conjugation) acts on the \( \ell \)-adic Tate module of E. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is ind...

We study the problem of inhomogeneous diophantine approximation under certain primality restrictions.

We establish a large sieve inequality for power moduli in ℤ[i], extending earlier work by Zhao and the first-named author on the large sieve for power moduli for the classical case of moduli in ℤ. Our method starts with a version of the large sieve for ℤ 2. We convert the resulting counting problem back into one for ℤ [i] which we then attack using...

We investigate three combinatorial problems considered by Erd\"os, Rivat, Sark\"ozy and Sch\"on regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements...

We establish a large sieve inequality for power moduli in $\mathbb{Z}[i]$.

In this paper, we establish a general version of the large sieve with additive characters for restricted sets of moduli in arbitrary dimension for function fields. From this, we derive function field versions for the large sieve in high dimensions and for power moduli.

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $\ell \not= p$, the Frobenius automorphism (unique up to conjugation) acts on the $\ell$-adic Tate module of $E$. The characteristic polynomial of the Frobenius automorphism is defined over $\mathbb{Z}$ and is independent of $\ell$. Its s...

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of $p\gamma$, $p$ running over the primes and $\gamma$ being a fixed irrational, for Gaussian primes.

We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. The novelty lies in establishing power savings instead of power of logarithm saving, as obtained in earlier works. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of t...

We establish a Polya-Vinogradov-type bound for finite periodic multipicative characters on the Gaussian integers.

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes.

We investigate the distribution of $p\theta$ modulo 1, where $\theta$ is a complex number which is not contained in $\mathbb{Q}(i)$, and $p$ runs over the Gaussian primes.

It is proved that Epstein's zeta-function $\zeta_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + i\gamma$ with $ T \leq \gamma \leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 4...

We prove a large sieve inequality for square norm moduli in Z[i].

We study the problem of Diophantine approximation on lines in $\mathbb{R}^d$
under certain primality restrictions.

Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the
error term in a smoothed version of an estimate for the density of elliptic
curves with square-free discriminant by T.D. Browning and the author. To
achieve this improvement, we elaborate on our methods for counting weighted
solutions of inhomogeneous cubic congruences to pow...

We study the problem of Diophantine approximation on lines in R^2 with prime
numerator and denominator.

Let F be an algebraically closed field of positive characteristic p. The
third author and Will Turner gave an explicit description of the extension
algebra of Weyl modules for GL_2(F). This, in particular, produced an explicit
basis. We examine this basis and use it to give upper and lower bounds for the
growth behaviour of the dimensions of Ext-gr...

We give an alternative proof of a recent result by T.D. Browning and A.
Haynes (arXiv:1204.6374v1) on multiplicative inverses in sequences of intervals
and improve this result under additional conditions on the spacing of these
intervals.

Improving and extending recent results of the author, we conditionally
estimate exponential sums with Dirichlet coefficients of L-functions, both over
all integers and over all primes in an interval. In particular, we establish
new conditional results on exponential sums with Hecke eigenvalues and squares
of Hecke eigenvalues over primes. We employ...

We investigate the average number of solutions of certain quadratic
congruences. As an application, we establish Manin's conjecture for a cubic
surface whose singularity type is A_5+A_1.

The dynamical Mertens ’ theorem describes asymptotics for the growth in the number of closed orbits in a dynamical system. We construct families of ergodic automorphisms of fixed entropy on compact connected groups with a continuum of growth rates on two different growth scales. This shows in particular that the space of all ergodic algebraic dynam...

Under the generalized Lindel\"of Hypothesis in the t- and q-aspects, we bound
exponential sums with coefficients of Dirichlet series belonging to a certain
class. We use these estimates to establish a conditional result on squares of
Hecke eigenvalues at Piatetski-Shapiro primes.

In this paper, we study the average of the Fourier coefficients of a
holomorphic cusp form for the full modular group at primes of the form
$[g(n)]$.

We investigate the density of integer solutions to certain binary
inhomogeneous quadratic congruences and use this information to detect almost
primes on a singular del Pezzo surface of degree 6.

We investigate the first and second moments of shifted convolutions of the generalised divisor function d3(n).

For given non-zero integers a,b,q we investigate the density of integer
solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this
to establish the Manin conjecture for a singular del Pezzo surface of degree 2
defined over the rationals and to examine the distribution of elliptic curves
with square-free discriminant.

We investigate various mean value problems involving order 3 primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large sieve-type result for order 3 (and 6) Dirichlet chara...

In this paper, we develop a conditional subconvexity bound for Godement-Jacquet $L$-functions associated with Maass forms for $SL(3,Z)$. Comment: A subtle mistake in Lemma 3 has been found and it does not appear that it can be easily fixed.

We obtain a new average results on the Lang-Trotter conjecture on elliptic curves.

Let $\lambda(n)$ be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant $C > 0$ depending on the cusp form and every fixed $c$ in the range $1 < c < 8/7$, the mean value of $\lambda(p)$ is $\ll \exp (-C \sqrt{\log N})$ as p runs over all (Piatetski-Shapiro) primes of the form...

Let E be an elliptic curve defined over the rational numbers and r a fixed integer. Using a probabilistic model consistent with the Chebotarev density theorem for the division fields of E and the Sato–Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to x which have Frobenius trace equal to r, where r...

We establish a result on the large sieve with square moduli. These bounds improve recent results by S. Baier [S. Baier, On the large sieve with sparse sets of moduli, J. Ramanujan Math. Soc. 21 (2006) 279–295] and L. Zhao [L. Zhao, Large sieve inequality for characters to square moduli, Acta Arith. 112 (3) (2004) 297–308].

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse–Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consi...

We obtain new average results on the conjectures of Lang-Trotter and Sato-Tate about elliptic curves.

This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.

We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The

For an elliptic curve $E$ over $\ratq$ and an integer $r$ let $\pi_E^r(x)$ be the number of primes $p\le x$ of good reduction such that the trace of the Frobenius morphism of $E/\fie_p$ equals $r$. We consider the quantity $\pi_E^r(x)$ on average over certain sets of elliptic curves. More in particular, we establish the following: If $A,B>x^{1/2+\e...

Suppose that 1/2 ≦ λ < 1. Balog and Harman proved that for any real θ there exist infinitely many primes p satisfying p
λ-θ < p-(1-λ)/2+ ε (with an asymptotic result). In the present paper we establish that for almost all θ in the interval 0 ≦ θ < 1 there exist infinitely many primes p such that {p
λ-θ} < p
-min{(2-λ)/6,(14-9λ)/32}+ε. Thus we obtai...

We obtain average results on the Sato-Tate conjecture for elliptic curves for small angles.

In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.

In this paper, we establish theorems of Bombieri-Vinogradov type and
Barban-Davenport-Halberstam type for sparse sets of moduli. As an application,
we prove that there exist infinitely many primes of the form $p=am^2+1$ such
that $a\leq p^{5/9+\epsilon}$.

We deal with the distribution of the fractional parts of $p^{\lambda}$, $p$ running over the prime numbers and $\lambda$ being a fixed real number lying in the interval $(0,1)$. Roughly speaking, we study the following question: Given a real $\theta$, how small may $\delta>0$ be choosen if we suppose that the number of primes $p\le N$ satisfying ${...

We prove that the conditions $\lambda<5/19$ and $L\le T^{1/2}$ in Theorems 3 and 4 of our recent paper "On the $p^{\lambda}$ problem" can be omitted.

We establish a result on the large sieve with square moduli. These bounds impro ve recent results by S. Baier(math.NT/0512228) and L. Zhao(math.NT/0508125).

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then use this result together with Fourier techniques to obtain large sieve bounds for the case when S consists of squares. These bounds improve a recent result by L. Zhao...

Suppose that a > 2. We prove that the number of positive integers q ≦ Q such that there exists a primitive character χ modulo q with χ (n) = 1 for all n ≦ (log Q)a
is O(Q
1/(1-a)+ε).

We establish a large sieve bound for expressions of the form
$$\sum\limits_{r=1}^R \left\vert \sum\limits_{M < n\le M+N} a_ne\left(\alpha_rf(n)\right)\right\vert^2,$$
where $f(x)=\alpha x^2+\beta x+\theta\in \mathbb{R}[x]$ is a quadratic polynomial with $\alpha>0$ and $\beta\ge 0$. We also consider the case when $f(x)=x^d$ with $d\in \mathbb{N}$, $...

Suppose that a > 1. By using a method of Linnik employing his Large Sieve one may derive the following result. The number of primitive Dirichlet
characters χ with conductor ≤Q such that χ(n) = 1 for all n ≤ (log Q)a is O(Q2/a+ε). We improve the exponent 2/a for a > 2 by using a refined version by Heath-Brown of the Halasz–Montgomery ‘Large Values m...

We give a short alternative proof using Heath-Brown's square sieve of a bound
of the author for the large sieve with square moduli.

In this paper we aim to generalize the results in Baier and Zhao and develop a general formula for large sieve with characters to powerful moduli that will be an improvement to the result of Zhao.

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then apply our result to the case when S consists of sqares. In this case we obtain an estimate which improves a recent result by L. Zhao.

In several papers A. Balog, G. Harman and the author studied the distribution of p λ (mod 1), where λ is a given real number lying in the interval (0,1) and p runs over the prime numbers. One of the main questions in these papers can be formulated in the following way: Let a real θ be given. For what fixed positive real numbers τ is it possible to...

We prove that the conditions $\lambda<5/19$ and $L\le T^{1/2}$ in
Theorems 3 and 4 of our recent paper "On the $p^{\lambda}$ problem" can
be omitted.

In the present paper we establish new mean value estimates for shifted

Balog and Harman proved that for any λ in the interval 1/2 ≤ λ < 1 and any real θ there are infinitely many primes p satisfying (with an asymptotic result). In the present paper we prove that for 59/85 = 0-694... < λ < 1 the above exponent -(1-λ)/2+ε may be replaced by - min{max{(35-22λ)/129, 1/7}, 5/18-λ/6}+ε. This result in particular contains t...

Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p+2d) with p+2d≤y, where d ∈ N is fixed and y → ∞. Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewood’s circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all d≤y/2.
In...

We deal with the distribution of the fractional parts of p λ , p running over the prime numbers and λ being a fixed real number lying in the interval (0,1). Roughly speaking, we study the following question: Given a real θ, how small may δ>0 be chosen if we suppose that the number of primes p≤N satisfying {p λ -θ}<δ is close to the expected one? We...

Let r be a positive integer and f 1 , ..., f r be distinct polynomials in Z[X]. If f 1 (n), ..., f r (n) are all prime for infinitely many n, then it is necessary that the polynomials f i are ir-reducible in Z[X], have positive leading coefficients, and no prime p divides all values of the product f 1 (n) · · · f r (n), as n runs over Z. Assuming t...

In this expository article, we discuss about an extremal problem in lattice point combinatorics which is related to several important combinatorial questions and is of similar nature to a problem considered by Harborth with an aim to generalize the Erd˝ os-Ginzburg-Ziv theorem in a particular direction.

We deal with the distribution of the fractional parts of p�, p running over the prime numbers andbeing a fixed real number lying in the interval (0,1). Roughly speaking, we study the following question: Given a real �, how small may � > 0 be choosen if we suppose that the number of primes p ≤ N satisfying n p� − � o < � is close to the expected one...

We derive some new estimates for sums over zeta zeros of Dirichlet polynomials. Using these estimates, we improve some results of G. Harman and the author on the distribution of p mod 1, p running over the prime numbers, for fixed real in the range 0 < 0.1031.

## Projects

Projects (3)